Scientific Notation Division Worksheet, Examples, and Rules

Dividing in Scientific Notation

When Dividing in Scientific Notation you divide the coefficients that are in the problem. Dividing in Scientific Notation is different then Adding or Subtracting because you do not have to make the exponents equal beforehand. When dividing the coefficients, you will subtract the exponents. After you divide, check your answer to make sure the coefficient is in between 1 and 10. If the coefficient is not in between 1 and 10, you must move the decimal to make it in between 1 and 10. You add to the exponent for each space that you moved the decimal to the left. You subtract from the exponent for each space that you moved the decimal to the right.

Scientific notation is a way of writing numbers that are too large or too small to be conveniently written in standard decimal notation. It is commonly used in mathematics, science, and engineering to express very large or very small numbers. Scientific notation involves writing a number as a coefficient multiplied by a power of 10, where the coefficient is a decimal number between 1 and 10 and the power of 10 is an integer.

Dividing scientific notation can be a challenging task for students who are not familiar with the rules of exponents. When dividing two numbers in scientific notation, it is important to remember to divide the coefficients and subtract the exponents. The procedure involves dividing the whole numbers separately from the bases, which gives the new coefficient in the scientific notation answer. The resulting quotient should be expressed in scientific notation with the same base as the original numbers. It is essential to use the correct formula and follow the rules to get accurate results.

Common Core Standard: 8.EE.A.3
Related Topics: Square Roots, Cube Roots, Irrational Numbers, Powers of 10, Scientific Notation Intro, Converting Numbers to Scientific Notation, Converting Numbers from Scientific Notation, Adding and Subtracting in Scientific Notation, Multiplying in Scientific Notation
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What is Scientific Notation?

Scientific notation is a way of expressing large or small numbers in a more concise and manageable form. It is also known as standard form or exponential notation. In scientific notation, a number is written as the product of a coefficient and a power of 10. The coefficient is a decimal number greater than or equal to 1 and less than 10. The power of 10 is an integer that represents the number of places the decimal point must be moved to get the original number.

For example, the number 6,000,000 can be written in scientific notation as 6 x 10^6. The coefficient is 6, which is greater than or equal to 1 and less than 10. The power of 10 is 6, which means the decimal point has to be moved 6 places to the left to get the original number.

Scientific notation is useful because it simplifies calculations with very large or very small numbers. It also makes it easier to compare numbers that have different orders of magnitude.

In scientific notation, the exponent represents the number of times 10 is multiplied by itself. For example, 10^2 means 10 multiplied by itself two times, which is 100. Similarly, 10^-2 means 1 divided by 10 multiplied by itself two times, which is 0.01.

The base in scientific notation is always 10. The coefficient can be a decimal number or an integer, and it can be positive or negative. If the coefficient is negative, the number is less than 1 and the power of 10 is negative.

Scientific notation is similar to engineering notation, which uses powers of 1000 instead of powers of 10. However, scientific notation is more widely used in science and engineering.

How to Divide Scientific Notation

Dividing scientific notation involves dividing the coefficients and the exponents of the notation. The process is straightforward and can be done manually or with the help of a calculator.

Manual Division

To divide two scientific notations with the same base, follow these steps:

1. Divide the whole numbers separately. This will give you the new coefficient in your scientific notation answer.
2. Divide the exponents separately. This will give you the new exponent in your scientific notation answer.
3. Combine the new coefficient and exponent to get the quotient in scientific notation.

For example, to divide 7.5 x 10^4 by 2.5 x 10^2, divide the coefficients 7.5 ÷ 2.5 = 3. Then divide the exponents 4 - 2 = 2. The quotient is 3 x 10^2.

Using a Calculator

Dividing scientific notation can also be done using a calculator. Many scientific calculators have a feature that allows for the direct input of scientific notation. To divide two scientific notations using a calculator, follow these steps:

1. Input the dividend and divisor in scientific notation.
2. Press the division button.
3. The calculator will display the quotient in scientific notation.

Summary

Dividing scientific notation involves dividing the coefficients and exponents separately. The process can be done manually or with the help of a calculator. When dividing with a calculator, input the dividend and divisor in scientific notation and press the division button to get the quotient.

Dividing Scientific Notation with Negative Exponents

When dividing numbers in scientific notation that have negative exponents, it is important to understand the rules of exponents. The rule of exponents states that when dividing two numbers with the same base, you can subtract the exponents.

For example, let’s say you want to divide 3.2 x 10^4 by 4 x 10^-2. First, you need to divide the coefficients, which in this case is 3.2 divided by 4, which equals 0.8. Then, you need to divide the powers of 10, which is 10^4 divided by 10^-2. To do this, you subtract the exponents, which gives you 10^6.

Therefore, the final answer is 0.8 x 10^6, which can be simplified to 8 x 10^5.

When dealing with negative exponents, it is important to remember that a negative exponent means the number is in the denominator of a fraction. For example, 10^-3 is the same as 1/10^3.

When dividing two numbers with negative exponents, the negative exponent can be changed to a positive exponent by moving the base to the opposite side of the fraction. For example, 5 x 10^-4 divided by 2 x 10^-6 can be rewritten as (5/2) x (10^-4/10^-6), which simplifies to 2.5 x 10^2.

In summary, when dividing numbers in scientific notation with negative exponents, it is important to follow the rules of exponents and remember that negative exponents indicate that the number is in the denominator of a fraction.

3 Simple Scientific Notation Division Examples

When dividing numbers in scientific notation, the first step is to divide the coefficients and then subtract the exponents of 10.

1. Divide the coefficients together.
2. Subtract the exponents of the power of ten from each other.
3. If the coefficient is in between 1 and 10 then you are done solving.
4. If the coefficient is not in between 1 and 10, then you must move the decimal either left or right to make the coefficient in between 1 and 10.
5. If you move the decimal left, you add to the exponent, if you move the decimal right, you subtract from the exponent.

Here are a few examples to illustrate how to divide numbers in scientific notation:

Example 1

Divide (4.5 × 10^5) by (3 × 10^2).

First, divide the coefficients: 4.5 ÷ 3 = 1.5.

Next, subtract the exponents of 10: 5 – 2 = 3.

Therefore, the quotient is 1.5 × 10^3.

Example 2

Divide (1.2 × 10^6) by (4 × 10^3).

First, divide the coefficients: 1.2 ÷ 4 = 0.3.

Next, subtract the exponents of 10: 6 – 3 = 3.

Therefore, the quotient is 0.3 × 10^3 or 3 × 10^2.

Example 3

Divide (5.6 × 10^-4) by (4 × 10^-2).

First, divide the coefficients: 5.6 ÷ 4 = 1.4.

Next, subtract the exponents of 10: -4 – (-2) = -2.

Therefore, the quotient is 1.4 × 10^-2.

It is important to note that calculators can be used to perform scientific notation division. Most scientific calculators have a key labeled “EE” or “EXP” which can be used to enter numbers in scientific notation. To divide two numbers in scientific notation, simply enter the dividend, press the division key, then enter the divisor, and finally press the equals key to get the quotient.

5 Quick Dividing in Scientific Notation Practice Problems

Converting to Scientific Notation

Scientific notation is a way to express very large or very small numbers in a compact form. It is a shorthand notation that uses powers of ten to represent numbers. In scientific notation, a number is written as the product of a decimal number and a power of ten. The decimal number is always greater than or equal to 1 and less than 10.

To convert a decimal number to scientific notation, you need to move the decimal point to the left or right until the number is between 1 and 10. The number of places you move the decimal point determines the power of ten. If you move the decimal point to the left, the power of ten is positive. If you move the decimal point to the right, the power of ten is negative.

For example, to convert the number 123,000 to scientific notation, you need to move the decimal point five places to the left. The result is 1.23 × 10^5. The number 0.0000456 can be converted to scientific notation by moving the decimal point four places to the right. The result is 4.56 × 10^-5.

When converting a decimal number to scientific notation, it is important to keep track of the number of decimal places you move the decimal point. This number determines the power of ten in the scientific notation.

In addition to decimal numbers, integers and decimal numbers can also be converted to scientific notation. When converting an integer to scientific notation, you can think of it as a decimal number with no decimal places. For example, the integer 500 can be written as 5 × 10^2 in scientific notation.

Overall, converting numbers to scientific notation is a useful skill that can make working with large or small numbers more manageable.

Scientific Notation Real Life Examples

Scientific notation is a tool that is commonly used in mathematics, science, and engineering to represent very large or very small numbers in a more manageable way. This notation is based on powers of ten and is written in the form of a number between 1 and 10 multiplied by a power of ten.

Scientific notation is used in many real-life situations, including:

• Astronomy: Astronomers use scientific notation to represent the distance between celestial objects. For example, the distance between the Earth and the Sun is approximately 93 million miles, which can be written in scientific notation as 9.3 x 10^7 miles.
• Chemistry: Chemists use scientific notation to represent the mass of atoms and molecules. For example, the mass of a proton is approximately 1.67 x 10^-27 kilograms.
• Engineering: Engineers use scientific notation to represent very large or very small quantities in a more manageable way. For example, the speed of light is approximately 299,792,458 meters per second, which can be written in scientific notation as 2.9979 x 10^8 m/s.
• Finance: Financial analysts use scientific notation to represent large sums of money. For example, the national debt of the United States is approximately $28 trillion, which can be written in scientific notation as 2.8 x 10^13 dollars.
• Medicine: Medical researchers use scientific notation to represent very small quantities, such as the concentration of a drug in the bloodstream. For example, the concentration of a drug in the bloodstream may be 0.00001 milligrams per liter, which can be written in scientific notation as 1 x 10^-5 mg/L.

These are just a few examples of how scientific notation is used in the real world. By using scientific notation, scientists and engineers can work with very large or very small numbers more easily and accurately.

Scientific Notation Division FAQ

How do you divide numbers in scientific notation?

To divide numbers in scientific notation, you need to follow a specific process. First, divide the coefficients, which are the numbers before the power of 10. Then, subtract the exponents of the powers of 10. Finally, simplify the result by converting it back to scientific notation if necessary.

What is the process for dividing scientific notation by a whole number?

To divide scientific notation by a whole number, you need to convert the whole number to scientific notation by adding a power of 10 equal to the number of digits in the whole number. Then, follow the process for dividing numbers in scientific notation as usual.

Can you provide an example of dividing scientific notation by a whole number?

Sure. Let’s say you want to divide 5.6 x 10^7 by 4. First, convert 4 to scientific notation by adding a power of 10 equal to the number of digits in 4, which is 1. So, 4 becomes 4 x 10^1. Then, divide 5.6 by 4 to get 1.4. Finally, subtract the exponents to get 10^6, so the final answer is 1.4 x 10^6.

What is the formula for dividing numbers in scientific notation?

The formula for dividing numbers in scientific notation is (a x 10^m) / (b x 10^n) = (a / b) x 10^(m – n), where a and b are the coefficients and m and n are the exponents of the powers of 10.

How do you use a scientific notation calculator to divide?

To use a scientific notation calculator to divide, enter the first number in scientific notation, then divide by the second number in scientific notation. The calculator should automatically simplify the result to scientific notation if necessary.

What are some practice problems for dividing numbers in scientific notation?

Here are some practice problems for dividing numbers in scientific notation:

1. (3 x 10^5) / (2 x 10^2)
2. (6 x 10^7) / (5 x 10^3)
3. (9 x 10^4) / (3 x 10^2)

What are the rules for scientific notation?

The rules for scientific notation are:

1. The coefficient must be greater than or equal to 1 and less than 10.
2. The exponent must be an integer.
3. The number must be expressed as a coefficient multiplied by a power of 10.

How do you write 540000000 in scientific notation?

To write 540000000 in scientific notation, move the decimal point to the left until there is only one digit to the left of the decimal point. Count the number of places you moved the decimal point and use that as the exponent of 10. In this case, you would move the decimal point 8 places to the left, so the scientific notation form is 5.4 x 10^8.

Scientific Notation Division Worksheet Video Explanation

Watch our free video on how to solve Dividing in Scientific Notation. This video shows how to solve problems that are on our free Divide Scientific Notation worksheet that you can get by submitting your email above.

Watch the free Divide Scientific Notation video on YouTube here: How to Divide Scientific Notation

Video Transcript:
This video is about how to dividing scientific notation. You can get the dividing scientific notation worksheet we use in this video for free by clicking on the link in the description below.

In order to show you how to divide scientific notation, we’re going to do a couple problems from our scientific notation division worksheet. When dividing in scientific notation you will take the coefficients of the numbers written in scientific notation and you will divide them. Then you will take the powers of 10 and you will subtract the exponents.

If you can think about when you divide exponents we know that when we divide exponents you subtract them so if you were just given the problem 10 to the seventh divided by 10 to the fifth, you would do 10 to the seventh minus 5. You would subtract the exponents and you would get 10 squared. This rule for exponents still applies to our numbers in scientific notation here. Even though they’re in scientific notation division, the rules for the exponent will still remain the same so you will still subtract these.

The first problem we are going to practice to show dividing with scientific notation is number 2. Number 2 gives us 4 point 2 times 10 to the seventh divided by 2 point one times 10 to the fifth. We’re going to take the coefficients which are 4 point 2 and 2 point 1 and we will divide them.  We have four point two divided by two point one that is going to be multiplied times our power of 10, which in this case is 10 to the seventh divided by 10 to the fifth. We already know that when we divide exponents we subtract them so we’re going to do 10 to the 7 minus 5. Four point two divided by two point one is two and then our power of 10 will be 10 to the 7 minus 5 which is also 2. Our solution to this problem will be 2 times 10 to the 2nd power. This is how to divide with scientific notation in an easy way.

The next problem is number 3. This problem is about how to divide scientific notation with different exponents. It gives us 4 times 10 to the 9th divided by 2 times 10 to the fourth. Once again when you divide in scientific notation you’re going to take your coefficients and your to divide them. We’re gonna do four divided by two times our power of 10 which will be ten to the ninth minus ten to the fourth, or just nine minus four because we’re dividing. We’re subtracting the exponents four divided by two is two and then 10 to the ninth minus four is nine minus 4 which is five. So our solution is two times ten to the fifth power. This is the easiest way to learn dividing numbers in scientific notation.

The last problem we’re going to do to show you how to divide with scientific notation is number six. This problem gives us nine point three times ten to the seventh divided by three times ten to the thirteenth. Once again we’ll take our coefficients which are nine point three and three and we will divide them. Nine point three divided by three we’re going to write times our power of ten, which in this case we’re gonna do 10 to the seventh minus 10 to the 13th or just seven minus 13.

When we divide nine point three divided by three it’s three point one times ten and then seven minus 13 is negative six. Our solution to this will be three point one times 10 to the negative sixth power. This has been three examples of division of scientific notation practice problems. You can try all the practice problems by downloading the free dividing numbers in scientific notation worksheet above.